The attainment set of the $\varphi$-envelope and genericity properties

2019 ◽  
Vol 124 (2) ◽  
pp. 203-246
Author(s):  
A. Cabot ◽  
A. Jourani ◽  
L. Thibault ◽  
D. Zagrodny
Keyword(s):  
Convex Hull ◽  
Closed Convex Hull

The attainment set of the $\varphi$-envelope of a function at a given point is investigated. The inclusion of the attainment set of the $\varphi$-envelope of the closed convex hull of a function into the attainment set of the function is preserved in sufficiently general settings to encompass the case $\varphi$ being a norm in a power not less than $1$. The non-emptiness of the attainment set is guaranteed on generic subsets of a given space, in several fundamental cases.

scholarly journals Some results on isometric composition operators on Lipschitz spaces

2021 ◽  
Author(s):  
Abraham Rueda Zoca
Keyword(s):  
Convex Hull ◽  
Composition Operator ◽  
Metric Spaces ◽  
Composition Operators ◽  
Extreme Points ◽  
Necessary Condition ◽  
Closed Convex Hull ◽  
Lipschitz Spaces ◽  
Complete Characterisation

AbstractGiven two metric spaces M and N we study, motivated by a question of N. Weaver, conditions under which a composition operator $$C_\phi :{\mathrm {Lip}}_0(M)\longrightarrow {\mathrm {Lip}}_0(N)$$ C ϕ : Lip 0 ( M ) ⟶ Lip 0 ( N ) is an isometry depending on the properties of $$\phi $$ ϕ . We obtain a complete characterisation of those operators $$C_\phi $$ C ϕ in terms of a property of the function $$\phi $$ ϕ in the case that $$B_{{\mathcal {F}}(M)}$$ B F ( M ) is the closed convex hull of its preserved extreme points. Also, we obtain necessary condition for $$C_\phi $$ C ϕ being an isometry in the case that M is geodesic.

Ε‎-Fr ´Echet Differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Simple Proof ◽  
Common Point ◽  
Lipschitz Functions ◽  
Closed Convex Hull ◽  
Uniformly Smooth ◽  
Fréchet Differentiability ◽  
Finite Set ◽  
Frechet Differentiability

This chapter treats results on ε‎-Fréchet differentiability of Lipschitz functions in asymptotically smooth spaces. These results are highly exceptional in the sense that they prove almost Frechet differentiability in some situations when we know that the closed convex hull of all (even almost) Fréchet derivatives may be strictly smaller than the closed convex hull of the Gâteaux derivatives. The chapter first presents a simple proof of an almost differentiability result for Lipschitz functions in asymptotically uniformly smooth spaces before discussing the notion of asymptotic uniform smoothness. It then proves that in an asymptotically smooth Banach space X, any finite set of real-valued Lipschitz functions on X has, for every ε‎ > 0, a common point of ε‎-Fréchet differentiability.

scholarly journals Doubly stochastic right multipliers

1984 ◽  
Vol 7 (3) ◽  
pp. 477-489
Author(s):  
Choo-Whan Kim
Keyword(s):  
Convex Hull ◽  
Compact Group ◽  
Compact Convex ◽  
Strong Operator Topology ◽  
Left Translation ◽  
Closed Convex Hull ◽  
Topological Isomorphism ◽  
Doubly Stochastic ◽  
Borel Measures ◽  
Translation Operators

LetP(G)be the set of normalized regular Borel measures on a compact groupG. LetDrbe the set of doubly stochastic (d.s.) measuresλonG×Gsuch thatλ(As×Bs)=λ(A×B), wheres∈G, andAandBare Borel subsets ofG. We show that there exists a bijectionμ↔λbetweenP(G)andDrsuch thatϕ−1=m⊗μ, wheremis normalized Haar measure onG, andϕ(x,y)=(x,xy−1)forx,y∈G. Further, we show that there exists a bijection betweenDrandMr, the set of d.s. right multipliers ofL1(G). It follows from these results that the mappingμ→Tμdefined byTμf=μ∗fis a topological isomorphism of the compact convex semigroupsP(G)andMr. It is shown thatMris the closed convex hull of left translation operators in the strong operator topology ofB[L2(G)].

scholarly journals Characterization algorithms for shift radix systems with finiteness property

2014 ◽  
Vol 11 (01) ◽  
pp. 211-232 ◽  
Author(s):  
Mario Weitzer
Keyword(s):  
Convex Hull ◽  
Convex Polyhedron ◽  
The Other ◽  
Closed Convex Hull ◽  
Connected Component ◽  
Complicated Structure ◽  
Finiteness Property ◽  
A Chain ◽  
Interior Points

For d ∈ ℕ and r ∈ ℝd, let τr : ℤd → ℤd, where τr(a) = (a2, …, ad, -⌊ra⌋) for a = (a1, …, ad), denote the (d-dimensional) shift radix system associated with r. τr is said to have the finiteness property if and only if all orbits of τr end up in (0, …, 0); the set of all corresponding r ∈ ℝd is denoted by [Formula: see text], whereas 𝒟d consists of those r ∈ ℝd for which all orbits are eventually periodic. [Formula: see text] has a very complicated structure even for d = 2. In the present paper, two algorithms are presented which allow the characterization of the intersection of [Formula: see text] and any closed convex hull of finitely many interior points of 𝒟d which is completely contained in the interior of 𝒟d. One of the algorithms is used to determine the structure of [Formula: see text] in a region considerably larger than previously possible, and to settle two questions on its topology: It is shown that [Formula: see text] is disconnected and that the largest connected component has non-trivial fundamental group. The other is the first algorithm characterizing [Formula: see text] in a given convex polyhedron which terminates for all inputs. Furthermore, several infinite families of "cutout polygons" are deduced settling the finiteness property for a chain of regions touching the boundary of 𝒟2.

What is the Subdifferential of the Closed Convex Hull of a Function?

1996 ◽  
Vol 27 (6) ◽  
pp. 1661-1679 ◽  
Author(s):  
J. Benoist ◽  
J.-B. Hiriart-Urruty
Keyword(s):  
Convex Hull ◽  
Closed Convex Hull

scholarly journals Delta- and Daugavet points in Banach spaces

2020 ◽  
Vol 63 (2) ◽  
pp. 475-496
Author(s):  
T. A. Abrahamsen ◽  
R. Haller ◽  
V. Lima ◽  
K. Pirk
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Closed Convex Hull ◽  
Müntz Spaces ◽  
Daugavet Property ◽  
Diameter Two Property

AbstractA Δ-point x of a Banach space is a norm-one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet point. A Banach space X has the Daugavet property if and only if every norm-one element is a Daugavet point. We show that Δ- and Daugavet points are the same in L1-spaces, in L1-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are Δ-points, but none of them are Daugavet points. We also study the property that the unit ball is the closed convex hull of its Δ-points. This gives rise to a new diameter-two property that we call the convex diametral diameter-two property. We show that all C(K) spaces, K infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums.

On the continuity of closed convex hull

1989 ◽  
Vol 18 (3) ◽  
pp. 297-299 ◽  
Author(s):  
Murat Sertel
Keyword(s):  
Convex Hull ◽  
Closed Convex Hull

On the Dixmier property of simple C*-algebras

1982 ◽  
Vol 91 (1) ◽  
pp. 75-78 ◽  
Author(s):  
Norbert Riedel
Keyword(s):  
Convex Hull ◽  
Von Neumann Algebras ◽  
Present Note ◽  
Linear Span ◽  
Closed Convex Hull ◽  
Von Neumann ◽  
Tracial State ◽  
C Algebras ◽  
Unital Simple

A unital C*-algebra is said to satisfy the Dixmier property if for each element x in the closed convex hull of all elements of the form u*xu, u being a unitary in , intersects the centre of ((2), 2·7). The von Neumann algebras and also some other classes of C*-algebras are known to satisfy the Dixmier property (cf. (2), (3), (4), (6)). If is a simple C*-algebra which satisfies the Dixmier property then has at most one tracial state. In (3) Archbold raised the question whether there exists a unital simple C*-algebra which has at most one tracial state without satisfying the Dixmier property. In the present note we characterize the unital simple C*-algebras with at most one tracial state in terms of a condition which is similar to the Dixmier property, but is in fact formally weaker in the framework of simple C*-algebras. This characterization relies on the method used by Pedersen in (5) in order to show that for a unital simple C*-algebra which has at most one tracial state and at least one non-trivial projection the linear span of all projections in is dense in As an application we characterize those unital simple C*-algebras with a unique tracial state which satisfy the Dixmier property.

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